Let X1X2Xn be an iidrandom variables from Uniform theta 12th

Let X1,X2,...,Xn be an i.i.d.random variables from Uniform (theta- 1/2,theta + 1/2), theta R. Show that x(1)+x(n)/2 is a MLE of theta. Is it a unique MLE of theta in this case? Show that x(1)+X(n)/2 is consistent for theta. (You can write P(|x(1)+X(n)/2-theta| GT )= P(|X(1)-(theta-1/2)+X(n)-(theta+1/2)GT2). Then use the inequality P(|X+ a +y+b|GT2 ) LE P(|X + a| GT )+ P(|y+b|GT ) for random variable X and Y and constants a and b.)

Solution